HPR3781: The Joule Thief
The Joule Thief is a delightfully simple circuit that can light an LED that requires 2V or more from a battery that is depleted to 1V or less. There are three components in addition to the battery and LED: an NPN transistor, a resistor and a double wound inductor. The key thing about the inductor is that the same core shares two windings but in opposite directions.
The explanation of how it works is simple enough, as long as you already understand how inductors and transistors work. In other words, it isn't simple at all! In short, the double wound inductor and transistor conspire to generate a transient high voltage spike and so turn LED on and off so rapidly that the human eye cannot perceive it.
The idea of such a circuit is approaching its centenary but the name itself is only around 20 years old. I recommend you check out Big Clive's web site as it was he who came up with the name "Joule Thief". I also recommend this video in which he constructs a circuit using some ninja-level soldering skills.